Matched pairs of Lie algebras and Rota-Baxter Lie algebras

TL;DR

The paper develops a unified framework linking matched pairs of Lie algebras with Rota-Baxter Lie algebras of weight $-1$, showing how a RB Lie algebra $(\mathfrak g,B)$ induces a canonical matched pair $(\mathfrak g_+,\mathfrak g_-,\rhd,\blacktriangleright)$ on $(\mathfrak g_+\oplus\mathfrak g_-,)$ and a bicrossed product decomposition $\mathfrak g_+\bowtie\mathfrak g_- = \mathfrak g_1 \oplus \mathfrak g_2$; the component $\mathfrak g_1$ carries an RB structure isomorphic to $(\mathfrak g,B)$ and $\mathfrak g_2$ acquires an RB structure as well. It then connects quadratic RB Lie algebras to Manin triples by producing a nondegenerate invariant form $S'$ on $\mathfrak g_+\oplus\mathfrak g_-$ and showing that this data forms a Manin triple that decomposes into two quadratic Lie algebras with induced RB structures. Extending to groups, the authors show that every RB group of weight $-1$ generates a matched pair of groups, with group projections giving rise to RB structures and decompose the group into components that mirror the Lie-theoretic decompositions. Together, these results illuminate operator forms of Yang-Baxter-type factorisations and dressing-like decompositions in both Lie algebra and group settings, and establish a coherent bridge between RB theory and classical factorisation theorems.

Abstract

In this paper, we investigate the relationship between matched pairs of Lie algebras and Rota-Baxter Lie algebras. First, we show that every Rota-Baxter Lie algebra $(\mathfrak{g},B)$ of weight $-1$ gives rise to a matched pair of Lie algebras $(\mathfrak{g}_+,\mathfrak{g}_-,\rhd,\bhd)$, and we prove that the bicrossed product Lie algebra decomposes as $\mathfrak{g}_+\bowtie\mathfrak{g}_-=\mathfrak{g}_1\oplus\mathfrak{g}_2$. Moreover, we establish a Rota-Baxter Lie algebra structure on $\mathfrak{g}_1$ which is isomorphic to $(\mathfrak{g},B)$ as a Rota-Baxter Lie algebra, and we endow $\mathfrak{g}_2$ with a Rota-Baxter Lie algebra structure. Then we study the connection between quadratic Rota-Baxter Lie algebras and Manin triples. We prove that every quadratic Rota-Baxter Lie algebra of weight $-1$ gives rise to a Manin triple, and we obtain a decomposition theorem for this Manin triple. Finally, we show that every Rota-Baxter group induces a matched pair of groups and investigate the internal structure of the induced matched pair of groups.

Theorems & Definitions (96)

• Lemma 2.1
• proof
• Lemma 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Proposition 2.7
• proof